
where \(\mathbf {A}=\mathbf {B}=\frac{1}{2}(V_s+V_{as})~\mathbf {I}\) , \(\mathbf {C}=\frac{1}{2}(V_s-V_{as})~\mathbf {Z}\) , \(t\)  denotes transpose,
\(\mathbf {I}=\begin{pmatrix} {\begin{matrix} 1 & 0 \\0 & 1 \end{matrix}} \end{pmatrix}\)  and
\(\mathbf {Z}=\begin{pmatrix} {\begin{matrix} 1 & 0 \\0 & -1 \end{matrix}} \end{pmatrix}\) . The symplectic eigenvalues can be determined by \(\sqrt{\frac{\Delta \pm \sqrt{\Delta ^2-4\textup {Det}\mathbf {V}_{ent}}}{2}}\) , where \(\Delta =\textup {Det}\mathbf {A}+\textup {Det}\mathbf {B}+2\textup {Det}\mathbf {C}\) . The positive partial transposition (PPT) criterion [1]} is applied to describe the entanglement of the EPR entangled state, which is a sufficient and necessary condition for a two-mode entanglement state with continuous variables. The PPT value can be determined by \(\sqrt{\frac{\Gamma -\sqrt{\Gamma ^2-4\textup {Det}\mathbf {V}_{ent}}}{2}}\) , where \(\Gamma =\textup {Det}\mathbf {A}+\textup {Det}\mathbf {B}-2\textup {Det}\mathbf {C}\) . When the PPT value is less than 1, the two mode quantum states are entangled [1]}.
